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Some key results on vectors and matrices (cf. sections 2.2, 2.3 and supplement 2A)

1 2

x x x       =       x ⋮

Vector (column vector):

n

x    

vector):

[ ]

1 2 n

x x x ′= x ⋯

Transposed

• f a vector

(row vector):

2 2 2 1 2 n

L x x x = + +

x

⋯

Length of a vector:

1

Multiplication by constant:

1 2 n

cx cx c cx       =       x ⋮

Unit vector in direction of x :

2 2 2 2 2 2 1 2 c n

L c x c x c x c L = + + =

x x

⋯

Length:

1

L−

x x

2

1 1 1 1 2 2 2 2 n n n n

x y x y x y x y x y x y +             +       + = + =             +       x y ⋮ ⋮ ⋮

Addition:

3

1 1 1 2 1 1 1 1 n i i i i k k i i i i i i n i i ni i

x c x x c c c x x

= = = =

            = =                

∑ ∑ ∑ ∑

x ⋮ ⋮

Linear combination:

1 k i i i

c

=

=

∑ x

Vectors x1,…., xn (of the same dimension) are linearly dependent if there exist constants c1,…. , cn , not all zero, such that Otherwise the vectors are linearly independent

4

Any set of n linearly independent vectors (of dimension n) is a basis for the vector space, and every vector may be expressed as a unique linear combination of a given basis Example with n = 4:

1 2 1 2 3 4 3 4

1 1 1 1 x x x x x x x x                               = = + + +                               x

5

Angle between vectors (n = 2)

1 1

cos( ) x L θ =

x 1 2

cos( ) y L θ =

y 2 1

sin( ) x L θ =

x 2

sin( ) y θ =

2 2

sin( ) y L θ =

y 2 1 2 1 2 1

cos( ) cos( ) cos( )cos( ) sin( )sin( ) θ θ θ θ θ θ θ = − = +

1 1 2 2 1 1 2 2 2 1

cos( ) cos( ) y x y x x y x y L L L L L L θ θ θ + = − = ⋅ + ⋅ =

y x y x x y

6

Inner product (n = 2)

1 1 2 2

cos( ) x y x y L L θ ′ + = = ′ ′ x y x x y y

1 1 2 2

x y x y ′ = + x y

2 2 1 2

L x x ′ = + =

x

x x

Length: Angle: L L ′ ′

x y

x x y y Remember that cos(90o) = cos(270o) = 0 The vectors x and y are perpendicular if their inner product is zero

7

Inner product (or dot product) in general:

1 1 2 2 n n

x y x y x y ′ = + + + x y ⋯

2 2 2 1 2 n

L x x x ′ = + + =

x

x x ⋯

Length: cos( ) L L θ ′ ′ = = ′ ′

x y

x y x y x x y y The vectors x and y are perpendicular (or orthogonal) if their inner product is zero Angle:

8

Projection of a vector x on a vector y : cos( ) L L L L L L L θ ′ ′ ′ ⋅ = ⋅ = ⋅ = ′

x x 2 y x y y y

y x y y x y x y y y y y

9

Matrix of dimension n x p :

11 12 1 21 22 2 1 2 p p n n np

a a a a a a a a a       =         A ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ Transposed matrix of dimension p x n :

10

Transposed matrix of dimension p x n :

11 21 1 12 22 2 1 2 n n p p np

a a a a a a a a a       ′ =         A ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯

Note: A column vector of dimension n is a n x 1 matrix A row vector dimension n is a 1 x n matrix

Multiplication by constant :

11 12 1 21 22 2 1 2 p p n n np

ca ca ca ca ca ca c ca ca ca       =         A ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯

11

Addition of matrices (of same dimension) :

11 11 12 12 1 1 21 21 22 22 2 2 1 1 2 2 p p p p n n n n np np

a b a b a b a b a b a b a b a b a b + + +     + + +   + =     + + +     A B ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ For a n x k matrix A and a k x p matrix B we have the matrix product

11 12 1 11 1 1 21 2 2 1 2 1 1 2 k j p j p i i ik k kj kp n n nk

a a a b b b b b b a a a b b b a a a               =                 AB ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ ⋯ ⋯

12

1 2 n n nk

a a a   ⋯

1 1 1 1 1 1 l l l lj l lp l l l il l il lj il lp l l l nl l nl lj nl lp l l l

a b a b a b a b a b a b a b a b a b         =          

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

⋯ ⋯ ⋮ ⋮ ⋮ ⋯ ⋯ ⋮ ⋮ ⋮ ⋯ ⋯

For a n x k matrix A and a k x n matrix B we have

( )′ ′ ′ = AB B A

A matrix A of dimension k x k is a square matrix If we say that the matrix is symmetric

′ = A A

13

The determinant of a k x k matrix A is denoted For a 2 x 2 matrix we have

11 22 12 21

a a a a = − A

For k larger than 2 we may use computer software

A

The rank of matrix equals the maximum number of linearly independent columns (or rows) Let A be a nonsingular k x k matrix. Then there exists a unique k x k matrix B such that A k x k matrix A is nonsingular if the columns of A are linearly independent, i.e. if the matrix has rank k

14

(the identity matrix) = = AB B A I

B is the inverse of A and is denoted by

1 −

A

For nonsingular square matrices A and B we have

1 1 1 1 1

( ) ( ) ( )

− − − − −

′ ′ = = A A AB B A

A k x k matrix Q is orthogonal if

1

• r

−

′ ′ ′ = = = QQ Q Q I Q Q

The columns of Q have unit length and are perpendicular (and similarly for the rows) Let A be a symmetric k x k matrix and x a k dimensional Let A be a symmetric k x k matrix and x a k dimensional vector

1 1 k k ij i j i j

a x x

= =

′ = ∑∑ x A x

is denoted a quadratic form

′ ≥ x A x

A is nonnegative definite provided that for all x

′ > x A x

A is positive definite provided that for all ≠

x

A k x k matrix A has eigenvalue λ with eigenvector if

λ = Ax x ≠ x

An eigenvalue λ is a solution to the equation

λ − = A I

A symmetric k x k matrix A has k pairs of eigenvalues and eigenvectors: The eigenvectors can be chosen to have length 1 and be mutually perpendicular. The eigenvectors are unique unless two or more eigenvalues are equal

1 1 2 2

, , ,

k k

λ λ λ e e e …

Here are the eigenvalues and are the associated perpendicular and normalized For a symmetric k x k matrix A we have the spectral decomposition

1 1 1 2 2 2 k k k

λ λ λ ′ ′ ′ = + + + A e e e e e e ⋯

1 2

, , ,

k

e e e …

1 2

, , ,

k

λ λ λ …

are the associated perpendicular and normalized eigenvectors (i.e.

1 and 0 for )

i i i j

i j ′ ′ = = ≠ e e e e